Nonlocality of single fermions branches that borrow particles

Transcription

1 1 Nonlocality of single fermions branches that borrow particles Sofia Wechsler Computers Engineering Center, Nahariya, P.O.B. 2004, 22265, Israel Abstract An experiment performed in 2002 by Sciarrino et al. provided a simple proof of the nonlocality of a single photon whose wave function is multi-branched. The difference between this experiment and others similar, is that the tester-particle used by Sciarrino to feel this nonlocality is another photon identical to the tested one. Such an experiment can be in principle performed with fermions too, and this is the case investigated in this article. The novel phenomenon revealed by Sciarrino s experiment is the particle borrowing. If a single particle is described by a two-branched wave function, then only one of these branches produces a detection at a time, the other branch remains silent. What happens in this experiment is that the silent branch borrows a particle from another source, if available in the neighborhood, and also produces a detection. To illustrate this feature more obviously, a modification of Sciarrino s experiment is proposed. Two sources of particles are made available in the neighborhood of the two branches. What then happens is that each branch takes and populates itself with a particle from whichever source is at hand. Keywords: Target particle, Tester particle, Particle borrowing Merge device 1. Introduction In 1967 Pfleegor and Mandel performed an experiment, [1], whose purpose was to test P.A. M. Dirac s famous dictum, [2], each photon interacts only with itself. Interference between different photons never occurs. Pfleegor and Mandel used two laser beams of so low total intensity that only one photon could be detected at a time. And they obtained interference. In their experiment, in the region where the two laser beams intersected, the source of the photon is not a constant of motion, it is undetermined. This type of indetermination is the topic of the present article, but this time with fermions, and with a more complex set of implications, as shown below. In order to prove the fact that two independent, low-intensity sources, one in the region A and one in the region B, can form a single-particle beam of the type 2 ½ ( 1> A + e iϕ 1> B ), i.e. a coherent superposition, one doesn t have to bring the two beams

2 2 to intersect. They may remain to travel each in its region, and their relative phase can be determined using local probes. S. M. Tan proposed to use coherent light as such local probes, [3]. His scheme for determining the relative phase between distant branches of a single-particle wave function, was implemented in many experiments, [4]. However coherent beams cannot be built with fermions, this scheme is restricted to bosons. Alternative types of probes were proposed, [5], [6], in the form of a superposition of the vacuum, 0>, and the Fock state 1>. In January 2002 Sciarrino et al. reported an experiment, [7], in which the probe particle is a single photon, (in the Fock state 1>), transformed by a beam-splitter into a superposition of two spatial modes, by means of which they tested the spatial modes of the target-photon. Such a type of test is however nonlocal, (see the discussion in [8]), but Sciarrino s experiment and conclusions didn t require local tests. Sciarrino s experiment revealed a couple of interesting features. First of all it confirmed very strikingly the nonlocal nature of single particles. Indeed, the organization of his experiment was so that two spatial modes, one from the target photon, one from the testing photon, fly to a third region in the space, to the experimenter Eve, and the two remaining modes, one from each photon, fly to a fourth experimenter, Victor. Within the class of detections in coincidence by Eve and Victor, an interference pattern was obtained, showing that although one of the two photons is detected by Eve, it is also present in Victor s lab, causing interference with the other photon detected there. Second, this experiment reveals a curious phenomenon, particles borrowing. When the wave function of a single particle comprises two branches, then only one of them can produce a detection at a time, the other branch remains silent. However, if a source of identical particles is neighbor, the silent branch can borrow from there a particle, the total number of particles in the system becomes 2, and that branch also can produce a click. And all that, while preserving the form of the wave function, the coefficients and relative phases in the superposition. To demonstrate this phenomenon, the present text proposes a modification of Sciarrino s experiment. Two local probes are introduced in Victor s lab. One in the left region where arrives the branch of target particle, one on the right where arrives the branch of the tester-particle. Each probe is merged with the neighbor branch of the target (tester), using for instance a beam-splitter. 1) A pair of outputs, one from each beam-splitter, is considered. The following effect is observed: if on the left is detected the target particle, the output on the right borrows a particle from the probe there. Alternatively, if on the right is detected the tester-particle, the output on the left borrows a particle from the neighbor probe. Both outputs produce click in the detectors, and the two-particle statistic shows the same interference pattern as if one would obtain if the target and tester particles intersected directly. An entanglement was formed between the two branches 2), the branch of the target on the left and the branch of the tester on the right. However each one of the branches 1) In fact Sciarrino et al. considered doing their experiment this way, but for experimental simplicity they preferred the scheme in [7]. 2) Duan discusses in [9] the difference between entanglements between modes and entanglements between particles.

3 3 takes and populates itself with whichever particle is at hand out of the four, target, tester, and probes. Sciarrino s experiment was performed with photons, but the present article proposes to work with fermions. That for the simple reason that most of the experiments are presently done with photons, and in order to confirm properties of the nature these have to be proved also on particles with rest-mass and also on fermions, not only on bosons. The mathematical treatment below is carried, as much as possible, without the entanglement-with-vacuum hypothesis used in Sciarino et al. paper. This hypothesis seems to me to have weaknesses. The rest of the text is organized as follows. Section 2 begins the description of a Sciarrino-type experiment, however with fermions. Section 3 presents a coherent superposition built with branches of different particles (different as source, not as type). Section 4 proposes a test for this coherence, adding two probes in Victor s lab. Section 5 discusses the phenomenon of particle borrowing. 2. A Sciarrino-type experiment The setup described in fig. 1 below is similar to that used by Sciarrino et al., however the discussion is about fermions, not photons. A fermion produced by the source A in the lab of the experimenter Alice, passes through a 50:50 beam-splitter, and two branches, a and c, emerge. An identical fermion produced by the source B in the lab of the experimenter Bob passes through an identical beam-splitter and generates the branches b and d. The branches a and b fly to South to a third party, Eve, while the branches c and d fly to North to a fourth party, Victor. In South, the branches a and b land on an additional 50:50 beam-splitter. The mirror M on the path b may be slightly displaced, and that introduces a phase shift ϕ. All the paths, a, b, c, d are equal in length. All the fermions in this experiment are prepared with spin up along the direction z, and are filtered so as to possess the same energy. Their wave packets are supposed to be tightly localized, to make possible to say whether a double detection is a coincidence, or these are two isolated events. For the purpose of this discrimination, the time axis is slotted into windows equal to the coherence time of a single fermion s wave packet. Two detections in a same time-window are considered a coincidence. Neither isolated detections, i.e. in different time-windows, nor detections of both particles in North or both in South, are relevant to us. We retain only cases of detections in coincidence, one in North and one in South. Let s now see how the things work. Assume that the description of the two fermions after exiting Alice s and Bob s beam-splitter, is (1) φ> A = 2 ½ ( a> + i c>), φ> B = 2 ½ ( b> + i d>).

4 4 Considering also the phase-shift ϕ on the path b, the total system state is (2) φ> = ½ {( e iϕ a> b> c> d>) + i( a> d> + e iϕ c> b>)}. As said above, only simultaneous detections, one in North, one in South are relevant here. These cases are comprised between the second pair of round parentheses in (2), (3) φ > = 2 ½ ( a> d> + e iϕ c> b>)}. VICTOR c d ALICE BOB a b A EVE ϕ M B e 1 e 2 E 1 BS E E 2 Figure 1. A Sciarrino-type experiment. The transformation in Eve s beam-splitter is (4) a> -> 2 ½ ( e 2 > A + i e 1 > A ), b> -> 2 ½ ( i e 2 > B + e 1 > B ), where the subscripts 1, 2, were given according to the detector number on the respective path, and A, B, remind the source of the fermion. Introducing (4) in (3) we get (5) φ > = ½ {( i e 1 > A d> + e iϕ e 1 > B c>) + ( e 2 > A d> + ie iϕ e 2 > B c>)}.

5 5 3. A coherent superposition from different sources? We refer below to the case of a click in the detector E 1. If the detector E 2 clicked, the rationale is completely analogous. The question we pose is whether there is any difference between the waves e 1 > A and e 1 > B, or one may write (6) e 1 > B = e 1 > A = e 1 >. Following a discussion of L. Mandel on such issues, [10], if there is an object or a property that remembers whether e 1 came from Alice s fermion or from Bob s fermion, then (6) is false. Eq. (5) shows that there is such an object, the particle going to North. We can place two detectors, one on the path c and one on the path d, and if the detector on c clicks then what is detected on e 1 is Bob s particle, or, if the detector on d clicks what is detected on e 1 is Alice s particle. But a detection on c even if it doesn t end with the absorption of the particle, destroys the branch d, and vice versa (the so called collapse ). We can t afford detectors in such positions. Could there be another object on which the origin of e 1 is engraved? L. Mandel suggested that the very sources A and B can, under some conditions, be such an object. But it s not our case, [11]. In general, an additional object or property to remember the origin of e 1 would mean an additional member in the entanglement, a third factor in the products in (3) and (5). That would mean a modification of our experiment and of the statistic we gather. No! We perform our experiment as written in (3) and (5), not otherwise! The last option worth checking is whether the detections at Victor s station can disclose the origin of e 1. Maybe they can tell us whether (6) is correct or not. Assume that (6) is correct and let s introduce it in (5). (7) φ > -> ½ { i e 1 >( d> e iϕ c>) + } That implies that when the detector E 1 clicks, the branches d and c remain in the coherent superposition (8) ψ> = 2 ½ ( d> ie iϕ c>). Please notice! The branches c and d belong to two different fermions, (different as source, the particles are of identical type), and traveling in space-separated regions. 4. A coherence test We are going to check if the coherent superposition (8) is indeed produced. The test proposed below, fig. 2, is more complicated than that in Sciarrino s paper, but the implications are more obvious. The branches c and d are left at a distance of one another, never meet, but each one of them is merged with a local probe, by means of a local beam-splitter.

6 6 In many cases in which the nonlocality of single photons is proved, the probes are coherent state beams, as proposed Tan, [3]. However, one cannot build coherent beams with fermions, (see for instance the discussion of Aharonov and Vaidman in [8]), because one cannot have two or more identical fermions in the same mode. We will adopt here the idea of Peres, [5], and later Lee et al., [6], and build our probes in the form (9) Q 1 > = β 0> + α q 1 >, Q 2 > = β 0> + α q 2 >, where 0> is the vacuum state, and α is very small, α << β. The total state of the probes is then (10) Q> = Q 1 > Q 2 > β 2 0> + α( q 1 > + q 2 >), where the term with two particles, α 2 q 1 > q 2 >, was neglected since its probability of occurrence is of the order of α 4. One can see that in Q> the term containing a single particle, is a non-local state, though, the probes are local and independent, satisfying the requiring of local tests discussed in [8]. The beam-splitters BS 1 and BS 2 perform a merge operation of the branches c, d, with the probe Q 1, respectively Q 2. The contribution of the inputs to the outputs v 1 and v 2 of the beam-splitters are as follows c> => 2 ½ v 1 > A d> => 2 ½ v 2 > B (11) (i), (ii). Q 1 > => 2 ½ i v 1 > Q1 Q 2 > => 2 ½ i v 2 > Q2 VICTOR V 1 V 2 v 1 v 2 Q 1 Q 2 BS 1 BS 2 c d Fig. 2. A test to verify the coherence of the modes c and d. after a detection in Eve s detectors E.

7 7 where the symbol => stands for contributes. Of course, there is loss of half intensity from each input, but this is not our concern here. In the absence of the probes the coherent superposition (8), if existent, should reappear at the output ports v 1 and v 2, at half intensity (12) ψ> => ½ ( v 2 > B ie iϕ v 1 > A ). With the probes in place, the total system entering Victor s beam-splitters has to be (13) In> 2 ½ { β 2 ( d> ie iϕ c>) 0> + α( q 2 > d> ie iϕ q 1 > c>) + α( q 1 > d> ie iϕ q 2 > c> )}. This expression leads to the following categories of outputs: *) a single detection either in the detector V 1 or in the detector V 2, 3) **) two detections in coincidence one in V 1 the other in V 2. We study the second category, comprised between the 3 rd pair of round parentheses. Introducing (11) in it, (14) Out double > = 2 3/2 iα( v 1 > Q1 v 2 > B ie iϕ v 2 > Q2 v 1 > A ). Now, if the following equalities similar with (6), hold, (15) v 1 > Q1 = v 1 > A = v 1 > A (i), v 2 > Q2 = v 1 > B = v 2 > (ii), one should get from (14) (16) Prob[v 1,v 2 ] ~ α 2 sin 2 (ϕ/2 + π/4). The experimental confirmation of (16) would prove the truth of (8). Indeed, if one blocks c and d, one has at the input ports of the beam-splitters only (10). There is no ϕ-dependence in (10), neither will be if one introduces in it (11), i.e. one looks at the output resulting from (10). It also can be easily checked that if one blocks only one of the branches c or d, still there is no ϕ-dependence. On the other hand, with all the inputs in place, one can see that (14) retains the same ϕ dependence as (8). 5. Borrowing particles But there is another, more interesting outcome of this test, the particle borrowing. Let s examine (12) again. As long as the total number of particles in the system is 1 either the branch v 2, or the branch v 1 can produce a click, not both. 3) It has to be noticed that in the category * ) fall also events, see the second pair of round parentheses in (13), in which actually two particles exit the same beam-splitter. However, even if we place detectors on all the output channels of BS 1 and BS 2 and obtain the probabilities of these events, they won t display any dependence on ϕ and won t show any signature of the coherent superposition (8).

8 8 As (14) shows, since additional sources of identical particles are available, a particle is borrowed. If the northern fermion (Bob s) produces a click on the branch v 2, the branch v 1 borrows a particle from the probe Q 1. If the northern fermion (Alice s) produces a click on v 1, the branch v 2 borrows a particle from the probe Q 2 with an amplitude of probability iα. The expression (14) is an entanglement between two modes v 1 and v 2, that populate themselves with whatever particle is at hand from the four sources, Alice s, Bob s, Q 1 or Q 2. While in Pfleegor and Mandel s experiment interference was produced with a single particle whose source is undetermined, out of two possibilities, here this interference is projected on two particles with undetermined sources out of four independent sources. Moreover, the interference is checked with local measurements. References [1] R. L. Fleegor and L. Mandel, Interference of Independent Photon Beams, Phys. Rev. Vol. 159, No. 5, page 1084, [2] P. A. M. Dirac, Quantum Mechanics, Oxford University Press, London 1958, 4 th ed., page 9. [3] S. M. Tan, D. F. Walls, and M. J. Colett, Nonlocality of a single photon, Phys. Rev. Lett., 66, 252, (1991); see also D. F. Walls and G. J. Milburn, Quantum Optics, page 273, Springer Study Edition, Springer-Verlag 1994, 2 nd printing [4] S. A. Babichev, J. Appel, and A. I. Lvovsky, Homodyne tomography characterization and nonlocality of a dual-mode optical qubit,quant-ph/ ; B. Hessmo, P. Usachev, H. Heydari, and G. Björk, An experimental demonstration of a single photon nonlocality, quant-ph/ , (2003); 1er. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, and S. Schiller, Quantum state reconstruction of the single-photon Fock state, quant-ph/ A. Kuzmich, I. A. Walmsley, and L. Mandel, Violation of Bell's Inequality by a Generalized Einstein-Podolsky-Rosen State Using Homodyne Detection, Phys. Rev. Lett. 85, page1349, (2000). [5] A. Peres, Nonlocal effects in Fock space, Phys. Rev. Lett, 76, 2205, (1996), (quant-ph/ ). [6] J. W. Lee, E. K. Lee, Y. W. Chung, H. W. Lee, and J. Kim, Quantum Cryptography Using Single Particle Entanglement, quant-ph/ , [7] F. Sciarrino, E. Lombardi, G. Milani, and F. DeMartini, Delayed Choice Entanglement Swapping with Vacuum One Photon Quantum States, Phys. Rev. A, 66, , (2002), quant-ph/ ; E. Lombardi, F. Sciarrino, S. Popescu, F. DeMartini, Teleportation of Entangled States of a Vacuum One Photon Qubit, quant-ph/ [8] Y. Aharonov and L. Vaidman, Nonlocal aspects of a quantum wave, Phys. Rev. A, 61, , (2000), quant-ph/ [9] L. M. Duan, J. I Cirac, and P. Zoller, Quantum entanglements in spinor BEC condensates, quant-ph/ [10] L. Mandel, Quantum effects in one-photon and two-photon interference, Reviews of Modern Physics, 71, no. 2, Centenary 1999, page S274. [11] Assume for simplicity that our fermions are electrons emitted from neutral atoms

9 9 that become ionized. Then we can learn from which source was emitted e 1 if we can establish in which source we got one more ionized atom than in the other. But the things aren t simple. One, because all the neutral atoms are identical and all the ionized atoms are identical. One cannot tell the newly ionized atom from all the others. Second, in each source there is a rich population of ionized atoms and a rich population of neutral atoms, and in each of them there is some uncertainty. One more atom that became ionized is less than this uncertainty and won t change the two populations.